The exactly solved 4D IQHE model in [1]}was isotropic in all four dimension and, in the presence of a flat boundary, the dispersion was found to display a Weyl singularity \(E(\vec{k}_\Vert ) = \pm \Vert \vec{k}_\Vert -\vec{k}_0\Vert \) , where \(\vec{k}_\Vert \)  are the three quasi-momenta parallel to the boundary and \(\vec{k}_0\)  is the coordinate of the Weyl singularity. Our acoustic crystal is highly isotropic and the Weyl singularity for a flat boundary is collapsed, in the sense that the boundary spectrum displays dispersion only with respect to \(\phi _1\)  if the boundary is cut perpendicular to the first spatial direction. Nevertheless, the Weyl physics of the boundary is still encoded (see [1]}in the boundary topological invariant supplied by the 3-dimensional winding number of the gap unitary operator \(U_G(\mathbf {\phi }) = e^{\imath 2 \pi g(D_{\mathbf {\phi }})}\) , where \(D_\phi \)  is the dynamical matrix of the crystal with a boundary and \(g\)  is any continuous real valued function taking 0/1 value above/below the bulk gap. By construction, the spectral decomposition of \(U_G-I\)  involves only the boundary modes that are spatially localized at the edges of the sample. Now, our sample has four edges instead of just a flat one, hence, for the sake of the argument, it is more appropriate to consider a disk-shaped sample of very large radius \(R\) . Then the winding number is computed using the variables \(\phi _1\) , \(\phi _2\)  and the momentum \(k_\Vert \)  parallel to boundary, with the latter treated with the real-space methods.[3]} Specifically, the derivation \(\partial _{k_\Vert }(\cdot )\)  is replaced by the commutator \(\imath [\cdot ,R \hat{\phi }_3]\)  and \(\int d k_\Vert (\cdot )\)  is replaced by \(\frac{1}{2\pi R}{\rm Tr}(\cdot )\) , where \(\hat{\phi }_3\)  is the operator corresponding to the polar angle in the plane of the sample. The radius \(R\)  cancels out and the 3D winding number takes the form
\(W_3(U_G) = \Lambda _3 \sum _{\sigma \in {\mathcal {S}}_3} (-1)^\sigma \int {\rm d}^3 \mathbf {\phi }\, \prod _j U_G^{-1} \partial _{\sigma _j}U_G,\) 
